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Asymptotic Properties of a Special Solution to the (3,4) String Equation

Nathan Hayford

TL;DR

This work analyzes a special solution to the $(3,4)$ string equation linked to the multicritical quartic $2$-matrix model. It demonstrates that, in a defined parameter domain, the associated isomonodromic $ au$-function admits a topological-type expansion and exists asymptotically for a fixed Stokes data set, with the leading structure governed by a quintic algebraic equation for a root $oldsymbol{oldsymbol{ ad}}$. A Deift–Zhou steepest-descent analysis of the 3×3 Riemann–Hilbert problem yields a global parametrix and a topological expansion for the $ au$-function, linking the string equation to a Painlevé I degeneration under two double-scaling limits. These limits produce tritronquée Painlevé I behavior, confirming a conjecture connecting the $(3,4)$- and $1$-matrix models and clarifying the RG-flow-like connection between critical points. The results also relate the $ au$-function to Painlevé I τ-functions in the critical regime, via explicit normalizations and Hamiltonian structures, and open avenues toward a full monodromy-map understanding between 2×2 and 3×3 isomonodromic systems.

Abstract

We analyze the asymptotic properties a special solution of the $(3,4)$ string equation, which appears in the study of the multicritical quartic $2$-matrix model. In particular, we show that in a certain parameter regime, the corresponding $τ$-function has an asymptotic expansion which is `topological' in nature. Consequently, we show that this solution to the string equation with a specific set of Stokes data exists, at least asymptotically. We also demonstrate that, along specific curves in the parameter space, this $τ$-function degenerates to the $τ$-function for a tritronquée solution of Painlevé I (which appears in the critical quartic $1$-matrix model), indicating that there is a `renormalization group flow' between these critical points. This confirms a conjecture from [1]. [1] The Ising model, the Yang-Lee edge singularity, and 2D quantum gravity, C. Crnković, P. Ginsparg, G. Moore. Phys. Lett. B 237 2 (1990)

Asymptotic Properties of a Special Solution to the (3,4) String Equation

TL;DR

This work analyzes a special solution to the string equation linked to the multicritical quartic -matrix model. It demonstrates that, in a defined parameter domain, the associated isomonodromic -function admits a topological-type expansion and exists asymptotically for a fixed Stokes data set, with the leading structure governed by a quintic algebraic equation for a root . A Deift–Zhou steepest-descent analysis of the 3×3 Riemann–Hilbert problem yields a global parametrix and a topological expansion for the -function, linking the string equation to a Painlevé I degeneration under two double-scaling limits. These limits produce tritronquée Painlevé I behavior, confirming a conjecture connecting the - and -matrix models and clarifying the RG-flow-like connection between critical points. The results also relate the -function to Painlevé I τ-functions in the critical regime, via explicit normalizations and Hamiltonian structures, and open avenues toward a full monodromy-map understanding between 2×2 and 3×3 isomonodromic systems.

Abstract

We analyze the asymptotic properties a special solution of the string equation, which appears in the study of the multicritical quartic -matrix model. In particular, we show that in a certain parameter regime, the corresponding -function has an asymptotic expansion which is `topological' in nature. Consequently, we show that this solution to the string equation with a specific set of Stokes data exists, at least asymptotically. We also demonstrate that, along specific curves in the parameter space, this -function degenerates to the -function for a tritronquée solution of Painlevé I (which appears in the critical quartic -matrix model), indicating that there is a `renormalization group flow' between these critical points. This confirms a conjecture from [1]. [1] The Ising model, the Yang-Lee edge singularity, and 2D quantum gravity, C. Crnković, P. Ginsparg, G. Moore. Phys. Lett. B 237 2 (1990)

Paper Structure

This paper contains 20 sections, 34 theorems, 296 equations, 8 figures.

Table of Contents

  1. Introduction.
  2. Isomonodromic $\tau$-function.
  3. Statement of results.
  4. Notations.
  5. Acknowledgments.
  6. Preliminary Transformations.
  7. Existence of solution for sufficiently large values of the parameters.
  8. Spectral curve and construction of the $g$-function.
  9. Deift-Zhou steepest descent.
  10. Proof of Theorems \ref{['MainTheorem1']} & \ref{['Theorem:topologicalexpansion']}.
  11. Painlevé I asymptotics.
  12. Construction of the modified spectral curve(s).
  13. Critical PI asymptotics: $\eta>0$.
  14. Critical PI asymptotics: $\eta<0$.
  15. Proof of Theorems \ref{['MainTheorem2']} & \ref{['MainTheorem3']}.
  16. ...and 5 more sections

Key Result

Theorem 1.1

For $(\eta,\mu,\nu) \in D$, the $\tau$-function for the $(3,4)$ string equation with Stokes data STOKES_TRUNCATED, as defined by tau-function-definition admits the $\hbar\to 0$ asymptotic expansion where $C(\hbar)$ is a constant independent of $\eta,\mu,\nu$, and $\varsigma$ is the unique solution to the $5^{th}$ order equation sigma-eq on $D$ which is specified in Definition Domain-D-Definition

Figures (8)

  • Figure 1.1: The Stokes lines $\Gamma_j$ for the Riemann-Hilbert problem for $\Psi(\zeta;t_5,t_2,t_1)$. Each of the Stokes sectors is bisected by an anti-Stokes line, depicted by a dashed line. All contours are oriented outwards from the origin. The anti-Stokes line $(-\infty,0]$ is labeled by $\mathop{\mathrm{\mathbb{R}}}\nolimits_-$. The Stokes matrix $S_k$ is the matrix associated to the parameter $s_k$; these parameters are not all independent, and must satisfy the equation $S_{-7}\cdots S_{-1}S_{1}\cdots S_{7} = \mathcal{S}^T$.
  • Figure 1.2: Critical surface in the $\eta,\mu,\nu$ -parameter space. Theorem \ref{['MainTheorem1']} holds for $(\eta,\mu,\nu)$ below this surface. On this surface, the two curves $\gamma_{-}$ and $\gamma_+$ are shown in orange and pink, respectively; the origin $(0,0,0)$ is depicted in red.
  • Figure 2.1: (a) The jumps of ${\bf Y}(\lambda;\eta,\mu,\nu | \hbar)$. (b) The new contours $\hat{\Gamma}_{5}$ (resp. $\hat{\Gamma}_{-3}$), and the new regions $\Delta_{\alpha}$ (resp. $\Delta_{\beta}$), which are enclosed by and the $\Gamma_5,\hat{\Gamma}_5$, and the real axis (resp. $\Gamma_{-3},\hat{\Gamma}_{-3}$, and the real axis). (c) The jumps of ${\bf Z}(\lambda;\eta,\mu,\nu | \hbar)$. In all figures, rays are oriented outwards.
  • Figure 3.1: (a) The sheets of the spectral curve $\mathcal{R}_j$, $j=1,2,3$, glued along the cuts $(-\infty,\beta]$ and $[\alpha,\infty)$. (b) The preimages of the sheets $\mathcal{R}_j$ in the uniformizing plane, with $a = 0.8,b = 3.2,c = 1.2$. The preimage of sheet $\mathcal{R}_j$ is labeled by its corresponding Roman numeral. Furthermore, there is the correspondence $\lambda(-a) = \alpha$, $\lambda(a) = \beta$. Dashed red lines correspond to the places where the $g$-function changes sign; we require that these curves do not intersect the branch cuts (shown in blue).
  • Figure 3.2: Final set of contours after lens opening for the Riemann-Hilbert problem for ${\bf R}$. All rays in the figure are oriented outwards; circles are oriented counterclockwise.
  • ...and 3 more figures

Theorems & Definitions (64)

  • Remark 1.1
  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.1
  • Theorem 1.4
  • Corollary 1.2
  • Remark 1.2
  • ...and 54 more